how to select a mesh with multiple faces quickly?
Usually Ctrl+L selects all linked vertices. From the image you posted it looks like this should be the case in your mesh. So, if that doesn't work it could be that your mesh has some issues. What I usually do in these cases is to select everything and then hit W  for the Specials Menu and then choose "Remove Doubles" in order to delete double vertices, which can occur if you have accidentally split or duplicated some faces. If you are lucky that fixes your mesh so that you can try the "Select Linked Vertices" again. 

Another thing you can try is "Select More": Ctrl+NumPad +.